Non-local operators are important observables in the Supersymmetric gauge theories in various dimensions. Line operators such as Wilson operators and 't Hooft operators are familiar examples. They serve as order parameters in the gauge theories that provide the valuable information about the phase structure in the non-perturbative regime. Similarly, Surface operators of Gukov and Witten [1]} with additional parameters
are expected to provide more information about the topological phases of gauge theories [2]}.
Surface operators in the 4d supersymmetric gauge theories are two-dimensional operators which may descend from the non-local operators in the six-dimensional \((0,2)\)  gauge theory, after suitable dimensional compactifications. These non-local operators in 6d theory can be either of codimension-2 or codimension-4 and they have been a focus of many investigations [3]}, [4]}, [5]}. Most of these studies have been done using the constructions in M-theory with intersecting \(M5\)  branes or \(M2\) -\(M5\)  branes. Recently, in [3]}, [4]} the analysis has used the probe M2 branes in the \(AdS_7 \times S^4\)  background to discuss a lot about the codimension-4 defects in the boundary \(6d\)  \(\mathcal {N}= (0,2)\)  theories, in terms of anomalies associated to the 2d surfaces that support them and from the bootstrap point of view. Whereas, [5]} discusses the defects of the both types and shows the equivalence between them in the IR regime, in the context of \(6d\)  \(\mathcal {N}= (0,2)\)  theories where the M5 worldvolumes wrap Reimann surfaces of non-trivial topology \(C\)  \(\subset \)  \(T^{\star }C\)  inside a hyper-Kähler manifold with the special holonomy group \(SU(2)\) . The M-theoretic construction of such type was first discussed in [9]} which was based on the eleven-dimensional uplift of type IIA models of Hanany and Witten in [10]}.
